# Implied Forward Rate in Exhibit 19 of CFAI Curriculum Reading 20

Could someone please help with this calculation in the Exibit 19 of CFAI Level 3 Reading 20.

I understand the calculation for implied forward rate (Column F) is explained in the notes below the exhibit. But that was for for the one-year forward rate at the end of year one (even that rate should be 2.32% instead of 2.33%, to be honest). I cannot get it right for 2.61% for Year 2 one-year forward rate (I believe the exhibt refers to one-year forward rate. But of course I can be wrong).

When I calculate the implied forward rate for Year 2, it seems my result is 2.87% and then the deviation gets bigger afterwards.

Can someone enlighten me a bit on this? Or is this implied forward rate really not derived from this calculation?

To find the implied forward yield you need to derive from “Bootstrapping” from the par curve as an input.

Assuming the ‘Derived Spot Interest Rate’ as:

Bond Maturity 1 Yr = 1.5%
Bond Maturity 2 Yr = 1.9139%
Bond Maturity 3 Yr = 2.2403%
Bond Maturity 4 Yr = 2.5191%
Bond Maturity 5 Yr = 2.7706%

Therefore, you can calculate ‘The Implied Forward Yields’ for Yr 1 as:

=> (1.019139)^2 = (1+0.015)^1 * (1+f)^1
=> f(1,1)= 2.33%

‘The Implied Forward Yields’ for Yr 2 as:

=> (1.022403)^3 = (1+0.015)^1 * (1+f)^2
=> f(1,2)= 2.61%

Fixed

=> (1.019139)^2 = (1+1.015)^1 x (1+f)^1
=> f(1,1)= 2.33%

‘The Implied Forward Yields’ for Yr 2 as:

=> (1.022403)^3 = (1+1.015)^1 x (1+f)^2
=> f(1,2)= 2.61%

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Thanks.

Many thanks guys.

Just 2 points more:

1. I assume the more precise par bond spot curve (such as 1.9139, 2.2403) is a widely accepted practice? (at least maybe I have seen it somewhere in L2. But in Reading 20, I cannot find the precise figure).
2. The forward rate we are talking to are actually f(1,2), f(1,3)… not f(1,1), f(2,1)… (actually I think f(1,2), f(1,3) may make more sense because it then use this to compare with Column H. But at first I cannot think it is a default assumption…)
1. For calculating forward yields, we need the spot rates, instead of the YTM of the bond, which is why bootstrapping is required to derive the spot rates from the par YTM. Bootstrapping is covered in L2 Fixed Income in the first reading “The Term Structure and Interest Rate Dynamics”. But you won’t need to do bootstrapping in Level 3.

2. In Column H, they just add 60 bps to the YTM of the bond. For example, for the 2-year bond, the column H indicates 2.51%, which indicates that one year from now, if you buy a 2-year zero-coupon bond, you can earn a yield of 2.51% per annum for 2 years (so holding period is 2 years).

Consequently, in column F, the implied forward yield has to be of a 2-year maturity so that it is comparable to column H for comparison. In this case the implied 2-year forward yield 1-year from now is 2.61%, which means that based on current yield curve, we should expect the 2-year spot rate/yield to be 2.61% per year for 2 years (in one year’s time). If you are using f(2,1) which is for a one-year zero coupon bond, then it would not be comparable to column H.

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In case, you want to know how the ‘Spot Interest Rates’ derive, you can calculate by using bootstrapping from the par curve:

• Spot rate Yr 1 =>
\begin{eqnarray} 100 &=& \frac{101.5}{(1+S_1)^1} \nonumber\\ S_1 &=& {1.5\%} \end{eqnarray}
• Spot rate Yr 2 =>
\begin{eqnarray} 100 &=& \frac {1.91}{(1+0.015)^1} + \frac {101.91}{(1+S_2)^2} \nonumber\\ S_2 &=& {1.9139\%} \end{eqnarray}
• Spot rate Yr 3 =>
\begin{eqnarray} 100 &=& \frac{2.23}{(1+0.015)^1} + \frac{2.23}{(1+0.019139)^2} + \frac{102.23}{(1+S_3)^3} \nonumber\\ S_3 &=& {2.2403\%} \end{eqnarray}

And so on…

• f(1,2) means starting one year from now for the implied two-year forward rate.

• f(2,1) means starting two years from now for the implied one-year forward rate.

Can you see the difference?

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Yes. Very clearly explained. Thanks.

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Many many thanks. And also…the very cool WORD-edited formula…

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